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Use the following example to answer the question.

Antoine goes to the mall to buy new shoes and jerseys. He pays a set rate per pair of shoes and a set rate per jersey, and he pays more per jersey than he pays per pair of shoes. His total spending can be modeled by the equation 35x+55y
. Which of the statements correctly interprets a part of the expression?

Statement #1: Antoine spends $35.00 per pair of shoes.

Statement #2: Antoine spends $55.00 per pair of shoes.

Statement #3: Antoine spends $35.00 in total for shoes.

(1 point)
Statement #
correctly interpre

11 answers

The correct interpretation of the expression \( 35x + 55y \) is provided in Statement #1.

Statement #1: Antoine spends $35.00 per pair of shoes.

This accurately reflects the coefficients of the variables in the equation, where \( x \) represents the number of pairs of shoes and \( 35 \) is the cost per pair of shoes.

The other statements are incorrect:

  • Statement #2 incorrectly states that Antoine spends $55.00 per pair of shoes; instead, $55.00 is the cost per jersey, not per pair of shoes.
  • Statement #3 incorrectly interprets the total spending; the expression does not indicate that Antoine spends a fixed total of $35.00 for shoes, but rather that he spends $35.00 for each pair he buys.
Consider the expression (16−12)2+5(8+22)53−(1+4)2
. What is the simplified numerator?(1 point)
Responses

76
76

42+60
4 squared plus 60

100
100

54

Let's simplify the expression step by step:

The expression is: \[ (16 - 12)^2 + 5(8 + 22)53 - (1 + 4)^2 \]

  1. Calculate \( (16 - 12)^2 \): \[ (16 - 12) = 4 \quad \Rightarrow \quad (4)^2 = 16 \]

  2. Calculate \( (8 + 22) \): \[ (8 + 22) = 30 \]

  3. Replace \( 5(30)53 \): Next, we substitute the values: \[ 5 \cdot 30 \cdot 53 \] To simplify, first calculate \( 5 \cdot 30 = 150 \): \[ 150 \cdot 53 \] Now calculate \( 150 \cdot 53 \): \[ 150 \cdot 53 = 7950 \]

  4. Calculate \( (1 + 4)^2 \): \[ (1 + 4) = 5 \quad \Rightarrow \quad (5)^2 = 25 \]

  5. Combine the results: The expression now looks like this: \[ 16 + 7950 - 25 \]

  6. Perform the final calculations: \[ 16 + 7950 = 7966 \] \[ 7966 - 25 = 7941 \]

Thus, the simplified numerator is: \[ \boxed{7941} \]

Since "7941" is not listed in your options, please double-check the expression you provided, as the options seem unrelated to the calculations done. Based on the arithmetic done here, the final answer does not match any of your provided responses.

Use the Commutative Property of Multiplication to rewrite the algebraic expression: k17⋅z33

The Commutative Property of Multiplication states that the order in which numbers are multiplied does not change the product. This means that we can rearrange the factors in a multiplication expression.

For the expression \( k^{17} \cdot z^{33} \), we can apply the Commutative Property of Multiplication as follows:

\[ k^{17} \cdot z^{33} = z^{33} \cdot k^{17} \]

So, the expression can be rewritten as: \[ z^{33} \cdot k^{17} \]

This is a valid application of the Commutative Property.

Which expression correctly demonstrates the Commutative Property of Multiplication?(1 point)
Responses

15(x+4)=15(x⋅4)
15 left parenthesis x plus 4 right parenthesis equals 15 left parenthesis x times 4 right parenthesis

15⋅x⋅4=4+15+x
15 times x times 4 equals 4 plus 15 plus x

15⋅(x)⋅4=4⋅15⋅(x)
15 times left parenthesis x right parenthesis times 4 equals 4 times 15 times left parenthesis x right parenthesis

x+4+15=15+4+x

To identify the expression that correctly demonstrates the Commutative Property of Multiplication, we need to find the one that shows that the order of multiplication does not affect the product.

The options provided are:

  1. \( 15(x+4) = 15(x \cdot 4) \)
  2. \( 15 \cdot x \cdot 4 = 4 + 15 + x \)
  3. \( 15 \cdot (x) \cdot 4 = 4 \cdot 15 \cdot (x) \)
  4. \( x + 4 + 15 = 15 + 4 + x \)

Analysis:

  • Option 1: This expresses an equation with addition inside the parentheses and does not demonstrate the commutative property
  • Option 2: This incorrectly states that multiplication is equal to addition.
  • Option 3: This properly shows that \( 15 \cdot (x) \cdot 4 \) can be rewritten as \( 4 \cdot 15 \cdot (x) \), demonstrating the Commutative Property of Multiplication by changing the order of the factors.
  • Option 4: This relates to addition, not multiplication.

Conclusion: The correct option that demonstrates the Commutative Property of Multiplication is:

Option 3: \( 15 \cdot (x) \cdot 4 = 4 \cdot 15 \cdot (x) \)

Write an equivalent expression to (a⋅b)⋅c
using the Associative Property of Multiplication.

The Associative Property of Multiplication states that the way in which factors are grouped in multiplication does not affect the product.

Given the expression \( (a \cdot b) \cdot c \), we can use the Associative Property to rearrange the grouping of the factors.

Using the Associative Property, the expression can be rewritten as: \[ a \cdot (b \cdot c) \]

So, an equivalent expression to \( (a \cdot b) \cdot c \) using the Associative Property of Multiplication is \( a \cdot (b \cdot c) \).

Which expression is equivalent to −3x−5
?(1 point)
Responses

(−4x−6)+(2+5x)
left parenthesis negative 4 x minus 6 right parenthesis plus left parenthesis 2 plus 5 x right parenthesis

−1+3x−(2x+6)
negative 1 plus 3 x minus left parenthesis 2 x plus 6 right parenthesis

−1+(x−4x)+6
negative 1 plus left parenthesis x minus 4 x right parenthesis plus 6

(x−7)+(2−4x)

To determine which expression is equivalent to \(-3x - 5\), we need to simplify each response option and see if it simplifies to \(-3x - 5\).

  1. Option 1: \((-4x - 6) + (2 + 5x)\) \[ = -4x - 6 + 2 + 5x \] Combine like terms: \[ = (-4x + 5x) + (-6 + 2) = 1x - 4 = x - 4 \quad \text{(not equivalent)} \]

  2. Option 2: \(-1 + 3x - (2x + 6)\) \[ = -1 + 3x - 2x - 6 \] Combine like terms: \[ = (3x - 2x) + (-1 - 6) = 1x - 7 = x - 7 \quad \text{(not equivalent)} \]

  3. Option 3: \(-1 + (x - 4x) + 6\) \[ = -1 + (1x - 4x) + 6 \] Simplify \(x - 4x\): \[ = -1 - 3x + 6 \] Combine like terms: \[ = (-3x) + (6 - 1) = -3x + 5 \quad \text{(not equivalent)} \]

  4. Option 4: \((x - 7) + (2 - 4x)\) \[ = x - 7 + 2 - 4x \] Combine like terms: \[ = (x - 4x) + (-7 + 2) = -3x - 5 \quad \text{(equivalent)} \]

Conclusion: The expression that is equivalent to \(-3x - 5\) is:

Option 4: \((x - 7) + (2 - 4x)\)